Coercivity of weighted Kohn Laplacians: The case of model monomial weights in ℂ²

Abstract

The weighted Kohn Laplacian ◻ φ \Box _\varphi is a natural second order elliptic operator associated to a weight φ : C n → R \varphi :\mathbb {C}^n\rightarrow \mathbb {R} and acting on ( 0 , 1 ) (0,1) -forms, which plays a key role in several questions of complex analysis. We consider here the case of model monomial weights in C 2 \mathbb {C}^2 , i.e., \[ φ ( z , w ) := ∑ ( α , β ) ∈ Γ | z α w β | 2 , \varphi (z,w):=\sum _{(\alpha ,\beta )\in \Gamma }|z^\alpha w^\beta |^2, \] where Γ ⊆ N 2 \Gamma \subseteq \mathbb {N}^2 is finite. Our goal is to prove coercivity estimates of the form \[ ( ∗ ) ◻ φ ≥ μ 2 , (*)\hspace {10pc} \Box _\varphi \geq \mu ^2,\hspace {10pc} \] where μ : C n → R \mu :\mathbb {C}^n\rightarrow \mathbb {R} acts by pointwise multiplication on ( 0 , 1 ) (0,1) -forms, and the inequality is in the sense of self-adjoint operators. We proved in 2015 how to derive from ( ∗ ) (*) new pointwise bounds for the weighted Bergman kernel associated to φ \varphi . Here we introduce a technique to establish ( ∗ ) (*) with \[ μ ( z , w ) = c ( 1 + | z | a + | w | b ) ( a , b ≥ 0 ) , \mu (z,w)=c(1+|z|^a+|w|^b) \qquad (a,b\geq 0), \] where a , b ≥ 0 a,b\geq 0 depend on (and are easily computable from) Γ \Gamma . As a corollary we also prove that, for a wide class of model monomial weights, the spectrum of ◻ φ \Box _\varphi is discrete if and only if the weight is not decoupled, i.e., Γ \Gamma contains at least a point ( α , β ) (\alpha ,\beta ) with α ≠ 0 ≠ β \alpha \neq 0\neq \beta . Our methods comprise a new holomorphic uncertainty principle and linear optimization arguments.